TITLE: Is the product of ideals commutative?
QUESTION [13 upvotes]: My algebra book introduces sum, intersection and product of ideals (in non-commutative rings), and then says that all three operations are commutative and associative, without proof.
I see no reasons why the product of ideals should be commutative, but I wasn't able to find a counterexample either.

REPLY [16 votes]: For another example, consider the ring $\mathbb{Z}\langle x,y\rangle$ be the ring of polynomials with integer coefficients in two noncommuting variables $x$ and $y$ (this is the same as the integral semigroup ring of the free monoid of rank $2$). It's almost like the usual polynomial ring, except that the monomials are given by words in $x$ and $y$; so that, for example, the weight two monomials are $x^2$, $xy$, $yx$, and $y^2$, all distinct; the weight three monomials are $x^3$, $x^2y$, $xyx$, $yx^2$, $xy^2$, $yxy$, $y^2x$, and $y^3$, all distinct; etc.
Let $I = (x)$, the principal ideal generated by $x$. It consists of all elements of the form
$$\sum_{i=1}^n p_i(x,y)xq_i(x,y)$$
with $p_i,q_i\in\mathbb{Z}\langle x,y\rangle$, $n\geq 0$; i.e., all polynomials in which every monomial has an $x$ in it, somewhere. Likewise, $J=(y)$ consists of all polynomials in which every monomial has a $y$ in it somewhere.
Now, clearly, $xy\in IJ$. But it cannot be in $JI$, because every monomial in every nonzero element of $JI$ must have an $x$ that follows its first $y$, which is not the case for $xy$.